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2
votes
1answer
128 views

Branch cuts in two-point function

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is ...
70
votes
0answers
3k views

Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
9
votes
0answers
270 views

Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory, when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In ...
7
votes
0answers
222 views

Dimensional regularization and IR divergences and scale invariance

I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant. Does dimensional regularization see "all" kinds of divergences? I mean - what ...
6
votes
0answers
117 views

Which values of the Riemann zeta funtion at negative arguments come up in physics?

For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. ...
6
votes
0answers
118 views

Srednicki's book chapter 8

Reading first page in chapter 8 of Srednicki's it reads: To employ the $\epsilon$ trick, we multiply $H_0$ with $1-i\epsilon$. The results are equivalent to replacing $m^2$ with $m^2-i\epsilon$. ...
5
votes
0answers
64 views

What is the probability of a Brownian path?

Suppose I have a Brownian particle with a diffusion constant $D$ starting out from a given position at time $0$ and follow it until time $\tau$. What is the probability (distribution) that it takes ...
4
votes
0answers
137 views

How does the renormalization scale $\mu$ cancel in all finite observables?

In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} ...
4
votes
0answers
99 views

can we PHYSCALLY (not by mathematics) justify that $ \zeta (-s)= 1+2^{s}+3^{s}+4^{s}+… $

the values $ \zeta (-1)= -1/12 $ and $ \zeta (-3)= 1/120 $ give accurate results for casimir and to evaluate the dimension in bosonic string theory so is there a physcial JUSTIFICATION to justify ...
4
votes
0answers
110 views

Cutoff regularization: Why not cutoff exactly at the momentum reached in an experiment?

So far I have only actually calculated dimensional regularization and I just know about the idea of cutoff regularization. From what I understand, as the name suggests, you just ignore momentum as ...
3
votes
0answers
104 views

Why do people rule out zeta regularization for renormalization?

Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. However, I have not seen anywhere. For example, I do not know why in ...
2
votes
0answers
86 views

Anomalies in QFT books

Why in most QFT books when author discusses of non-invariance of measure of path integral (massless fermions interact with gauge fields) $$ \int D\bar{\Psi} D\Psi \to |\Psi \to U\Psi , \quad ...
2
votes
0answers
51 views

renormalization by differentiation how does it work?

i mean let be the integral $$ \int_{0}^{\infty} \frac{p^{3}}{(p^{2}+m^{2})^{2}} $$ logartihmic divergence but if a apply differentiation with respect to $ m^{2} $ i get $$ \int_{0}^{\infty} ...
2
votes
0answers
31 views

Observable which dependes on the cutoff

In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: Where $\mu$ is the ...
2
votes
0answers
89 views

Cancellation of the quadratic divergence in QCD

I am currently reading about QCD in QFT-Peskin&Schroeder. When calculating 1-loop diagrams for QCD and using dimensional regularization, the 3-vertex boson loop, 4-vertex boson loop and ghost loop ...
2
votes
0answers
111 views

Regulating the sum in Casimir Force

I am trying to evaluate the Casimir force using a Gaussian regulator (I know there are other much easier ways to do this, but I want to try this!) We then are reduced to evaluating the sum $$ ...
2
votes
0answers
205 views

Regularization of infinite series: an alternative for the not-always-courteous-Zeta

Zeta-function regularization of infinite series is the most commonly used in QFT applications. However, occasionally other schemes are employed which, allegedly, suit the nature (most noticeably the ...
2
votes
0answers
152 views

Functional determinant approximation

Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$. I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use $$\sum_{n} ...
1
vote
0answers
39 views

Apparent elimination of overlapping divergences

The integral, $$ \iint_{\mathbb{R}^{2+}}\frac{xy}{1+x+y} \mathrm{d}y \, \mathrm{d}x$$ possesses an overlapping divergence when $ x \to \infty $ and $ y \to \infty $. However, under a change of ...
1
vote
0answers
19 views

What properties does the conductor making the plates for Casimir effect have?

Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM ...
1
vote
0answers
35 views

Regulating a particular function

I am interested in computing the integral of this function: \begin{align} \int_0^\infty\frac{2du(u^2+1)}{(1-e^{2\pi u})}, \end{align} which of course at first sight, does not converge. But in QFT ...
1
vote
0answers
96 views

How should I regularize this integral?

I need to calculate the following integral (which is divergent): \begin{equation} I(m,C)=\int_{-\infty}^\infty {\rm d}\omega\int_{\rm space}{\rm d^3 ...
1
vote
0answers
63 views

Sharp cut-off, quadratic corrections and naturalness

When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to ...
1
vote
0answers
110 views

IR divergence and renormalization scale in dimensional regularization (part 2)

This is in continuation of my previous question, IR divergence and renormalization scale in dimensional regularization. Lubos gave a nice answer there but I want to get to a very specific example ...
1
vote
0answers
42 views

does this expression appear in renormalization?

my questiion is if this regularizatio for the Harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} = \frac{ -\Gamma ' (a)}{\Gamma (a)}$$ for any positive and finite 'a' appears in renormalization ...
1
vote
0answers
160 views

Why does renormalization need an unbroken symmetry?

Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that ...
0
votes
0answers
46 views

how must i understand this 2-loop integral?

let be the 2-loop integral... $$ \int d^{d}l\int d^{d}k \frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \int ...
0
votes
0answers
191 views

$\mathrm{i}\epsilon$ prescription makes a function analytical?

I've seen this everywhere where they say "Analytic continuation is obtained by the usual $\mathrm{i}\epsilon$ prescription..." but how is that? How do you analytically continue (say) $\ln x$ with ...